In SageMath, you can enter the following:

U = CyclotomicField(43)
P.<x> = PolynomialRing(U)

def c(i):
    return U(cos(i*pi/43))

x1 = 2*(c(2) + c(12) + c(14))
x2 = 2*(c(6) + c(36) + c(42))
x3 = 2*(c(18) + c(22) + c(40))
x4 = 2*(c(20) + c(32) + c(34))
x5 = 2*(c(10) + c(16) + c(26))
x6 = 2*(c(8) + c(30) + c(38))
x7 = 2*(c(4) + c(24) + c(28))

(x-x1)*(x-x2)*(x-x3)*(x-x4)*(x-x5)*(x-x6)*(x-x7)

And you get:

x^7 + x^6 - 18*x^5 - 35*x^4 + 38*x^3 + 104*x^2 + 7*x - 49

that is: $x^{7} + x^{6} - 18 x^{5} - 35 x^{4} + 38 x^{3} + 104 x^{2} + 7 x - 49$.

Note that I cannot guarantee that the computations are exact; I don't quite know what the cos function does.

In SageMath, you can enter the following:

U.<zeta> = CyclotomicField(43)
P.<x> = PolynomialRing(U)

def c(j):  # cos(j * pi / 43)
    return (zeta ** j + zeta ** (-j))/2

x1 = 2*(c(2) + c(12) + c(14))
x2 = 2*(c(6) + c(36) + c(42))
x3 = 2*(c(18) + c(22) + c(40))
x4 = 2*(c(20) + c(32) + c(34))
x5 = 2*(c(10) + c(16) + c(26))
x6 = 2*(c(8) + c(30) + c(38))
x7 = 2*(c(4) + c(24) + c(28))

(x-x1)*(x-x2)*(x-x3)*(x-x4)*(x-x5)*(x-x6)*(x-x7)

And you get:

x^7 + x^6 - 18*x^5 - 35*x^4 + 38*x^3 + 104*x^2 + 7*x - 49

that is: $x^{7} + x^{6} - 18 x^{5} - 35 x^{4} + 38 x^{3} + 104 x^{2} + 7 x - 49$.